Two Player Zero-Sum Games Calculating Saddlepoints an algebraic approach

Finding the Saddlepoint b S4 c d Given any two-player zero-sum game, can we construct an algebraic formula to determine the saddle point?

Finding the Saddlepoint b S4 c d Given any two-player zero-sum game, can we construct an algebraic formula to determine the saddle point? Let’s assume that the saddle point does not occur in pure strategies. That is, assume the saddle point lies in mixed strategies…

Finding the Saddlepoint p 1-p Given any two-player zero-sum game, can we construct a formula to determine the saddle point in mixed strategies? The answer is yes. This is based on the assumption that the saddlepoint in mixed strategies will occur at the intersection of payoff functions defined as follows. S1 S2 S3 a b S4 c d q 1-q We will assign probability p to strategy S1 and thus 1-p to strategy S2 and likewise, we’ll assign probability q to strategy S3 and thus 1-q to strategy S4. Now the payoff functions for each player are as follows… ES1 = aq + c(1-q) = aq + c – cq = q(a-c) + c ES2 = bq + d(1-q) = bq + d – dq = q(b-d) + d ES3 = ap + b(1-p) = ap + b – bp = p(a-b) + b ES4 = cp + d(1-p) = cp + d – dp = p(c-d) + d

Finding the Saddlepoint p 1-p S1 S2 S3 a b S4 c d The optimal strategy for the column player will occur at the intersection of these two payoff functions: ES3 = p(a-b) + b ES4 = p(c-d) + d q 1-q …and the optimal strategy for the row player will occur at the intersection of these payoff functions. ES1 = q(a-c) + c ES2 = q(b-d) + d

Finding the Saddlepoint p 1-p ES3 = p(a-b) + b ES4 = p(c-d) + d Solving ES3 = ES4 for p we get… S1 S2 S3 a b S4 c d Note that this is undefined if the denominator is equal to zero.

Finding the Saddlepoint ES1 = q(a-c) + c ES2 = q(b-d) + d Likewise, solving ES1 = ES2 for q we get… S1 S2 S3 a b S4 c d q 1-q Again, this is undefined if the denominator is equal to zero.

Finding the Saddlepoint Given any two-player zero-sum game, can we construct an algebraic formula to determine the saddle point in mixed strategies? The answer is yes. We can use the following formulas to find p and q which determine optimal mixed strategies for each player… S1 S2 S3 a b S4 c d Of course, once p and q are determined, then we may calculate 1-p and 1-q and thus find optimal mixed strategies for each player. Note – these formulas are valid only when the optimal strategy for each is a mixed strategy and when (a+d) – (b+c) is not 0.

A B X -2 5 Y 7 1 You can try these formulas on the following example. A good idea is to try both methods – algebraic and geometric. That is, use the shortcut algebra formula but also try finding the saddle point by graphing the payoff functions. player 1 You will get the following answer: Optimal strategies are Player 1: (4/13, 9/13) Player 2: (6/13, 7/13) The value of the game is: 37/13 A B X -2 5 Y 7 1 player 2